Problem 41 - Thomas Calculus

Question

In Exercises \(37-42,\) write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=\pi r^{2} h\) of a right circular cylinder when the radius changes from \(r_{0}\) to \(r_{0}+d r\) and the height does not change

Step-by-Step Solution

Answer

\( dV = 2\pi r h \, dr \) for small changes in radius.

Step 1: Understand the given formula

The volume of a right circular cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder.

Step 2: Identify the variables and changes

We need to determine the change in volume \( dV \) when the radius changes from \( r_0 \) to \( r_0 + dr \) while the height \( h \) remains constant. Here, \( dr \) represents the small change in the radius.

Step 3: Differentiate the volume formula

To find \( dV \), differentiate the volume formula with respect to \( r \). Using calculus, we get the differential \( dV = \frac{d}{dr}(\pi r^2 h) dr = 2\pi r h \, dr \).

Step 4: Write the differential formula

The differential formula that estimates the change in the volume of the cylinder, given a small change in the radius \( dr \) and constant height, is \( dV = 2\pi r h \, dr \).

Key Concepts

Cylinder Volume Change

When we consider the volume of a cylinder, we focus on its two main dimensions: the radius and the height. The formula to calculate the volume of a cylinder is given by \( V = \pi r^2 h \) where:

• \( r \) is the radius of the cylinder's base
• \( h \) is the height

Analyzing the volume change primarily involves understanding how changes in these dimensions affect the volume. In this exercise, we keep the height constant and let the radius increase slightly by \( dr \).
The task then is to estimate how much this small change \( dr \) in the radius affects the overall volume of the cylinder. By calculating the differential \( dV \), we can approximate how changes impact the volume more effectively without recalculating every time from scratch. It becomes a handy tool especially when changes are minimal and direct measurement may not be feasible.

Differentiation

Differentiation is a core concept in calculus that allows us to understand how a function changes. It helps find the rate at which one quantity changes relative to another. When applied to the volume of a cylinder, it helps us determine how a tiny change in the radius affects the volume.To differentiate the volume formula of a cylinder, \( V = \pi r^2 h \), with respect to \( r \), we apply the power rule of differentiation:

• Differentiate \( r^2 \) which gives \( 2r \)
• Multiply it by \( \pi h \) since \( h \) is constant, leading to \( 2\pi r h \)

This differentiation essentially breaks down the volume formula into its rate of change, focusing on how the radius impacts the volume. This specific rate, \( 2\pi r h \), tells us the immediate impact on volume with respect to a change in radius.

Differential Formula

The differential formula provides an approximation of how a small change affects a function. In the case of our cylinder, the change in volume due to a slight radius change can be described using the differential formula.After differentiating the volume \( V = \pi r^2 h \), we derived that:\[ dV = 2\pi r h \, dr \]This expression helps estimate the change in volume (\( dV \)) when the radius of the cylinder is incremented by a small amount, \( dr \). The term \( 2\pi r h \) represents how sensitive the volume is to changes in the radius while holding the height constant.This formula is exceptionally useful for engineers and scientists to quickly assess potential impacts of minor changes in physical dimensions on volume, without the need to perform complex calculations repeatedly. It simplifies decision-making where precision can vastly affect outcomes in fields like manufacturing and construction.